Question 176379
A finite set with a smallest element could be  the positive numbers (1,2,3 etc)
   but then even adding a largest element does not necessarily  make it finite.
For example  the set of all Rational numbers(fractions in other words)   between 0 and 1. You would get ( 1/2 , 1/3,  1/4 , 1/5 etc)  again an infinite set.  A truly finite set can be listed individually.

Hope That helps!